Conduction of heat is the process where vibrating particles pass on their EXTRA KINETIC ENERGY to neighbouring particles.

This process continues throughout the solid and gradually, some of the extra kinetic energy/ heat is passed all the way through the solid,causing a rise in the temperature at the other side of the solid. And hence an increase in the heat radiating from its surface.

Metals conduct heat so well because the electrons are free to move inside the metal.

At the hot end, the electrons move faster and collide with the other free electrons, transferring energy. These other electrons then pass on their extra kinetic energy to other electrons.

Because the electrons are free to move, this is obviously a much faster way of transferring the energy through the metal, than slowly passing it between jostling neighbouring atoms.